For more information about this meeting, contact Becky Halpenny, Robin Enderle.
| Title: | "The Generalized Finite Element Method: Numerical Treatment of Singularities, Interfaces, and Boundary Conditions" |
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| Seminar: | Ph.D. Thesis Defense |
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| Speaker: | Qingqin Qu, Advisers: Anna Mazzucato/Victor Nistor, Penn State |
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| Abstract: |
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| This dissertation is devoted to use the Generalized Finite Element Method
to solve partial di erential equations numerically. As an extension of the
standard Finite Element Method (FEM), the Generalized Finite Element
Method (GFEM) is especially convenient for dealing with complicated do-
mains, corner singularities, transmission problems and mixed boundary con-
ditions. The GFEM is related to other methods, such as the hp cloud method
and the extended nite element method. The GFEM di ers from the stan-
dard FEM in the construction of the nite-dimensional space in which the
approximate solution is sought. Instead of using piecewise polynomials on
each element of a triangulation of the domain, we de ne the element spaces
by using partitions of unity to combine the local approximation spaces, de-
ned in each patch of the partition, in order to obtain the global GFEM
space. In particular, the GFEM is an example of a meshless method, since
the partition of unity need not be subordinated to a particular mesh. The
GFEM allows one to include a priori knowledge about the local behavior of
the solution, and gives the option of constructing trial spaces of any desired
regularity.
For transmission (interface) problems on domains with smooth, curved
boundaries, we establish quasi-optimal rates of convergence of the numerical
solution to the true solution by using a non-conforming Generalized Finite
Element Method. A sequence of approximation spaces Sn are constructed
that satisfy the following two conditions: (1) nearly zero boundary and inter-
face matching, (2) approximability. The numerical solution is then obtained as a Galerkin approximation to the true solution. We prove that the ap-
proximation error of order O(dim(Sn)
m=2
), where dim(Sn) is the dimension
of the GFEM space Sn, and m is the degree of polynomials used for the
local approximation of the solution. Numerical experiments are presented to
demonstrate these theoretical results.
For the case of singular domains, we study the GFEM approximation
to solutions of the Poisson's problem in polygons. It is well-known that
the loss of regularity of the exact solution due to domain singularities will
deteriorate the convergence rate of the standard FEM if one uses quasi-
uniform meshes. To circumvent the loss of regularity, we pose the problem in
certain weighted Sobolev spaces, and show that the continuous problem has
the expected regularity in these spaces. We construct GFEM approximation
spaces using again a partition of unit and local approximation spaces. For
the former, we use dilation techniques to deal with corner singularities, while
we use standard piecewise polynomial spaces for the latter. We then establish
quasi-optimal rate of convergence of the GFEM approximation to the exact
solution both in weighted Sobolev spaces and then in Hilbert spaces. |
Room Reservation Information
| Room Number: | MB106 |
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| Date: | 05 / 15 / 2012 |
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| Time: | 02:00pm - 04:00pm |
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